Overlapping Decomposition of
Load Flow Jacobian for Static
Voltage Stability Indicator in
Interconnected Power System
Marija Ilic & Le Xie
Dept. of ECE
Carnegie Mellon University
Problem Posing
z Needs for monitoring the interconnection
based on QIs essential for deciding the
severity of the operating mode in a
decentralized way
z Static voltage stability is an important and
starting point for research in power system
stability
z How to monitor static voltage stability from
some practically effective and decentralized
QIs?
1/10/2006
Ilic & Xie, ECE/CMU
2
Problem Solving: Model Review
z DAE Equations-> ODE Equations->
Linearized Model
– Monitoring load flow Jacobian determinant
can detect a possible dynamic instability
under certain assumptions [1]
z Load level producing zero determinant
can serve as an upper bound of steadystate stability
z Load flow Jacobian determinant is a main
QI for static voltage stability
1/10/2006
Ilic & Xie, ECE/CMU
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Decomposition of Load Flow Jacobian
z Properties of load flow Jacobian
Area I
Area II
⎡ J I −I
⎢J
J = ⎢ II − I
⎢ J III − I
⎢
⎣ J IV − I
J I − II
J II − II
J III − II
J IV − II
J I − III
J II − III
J III − III
J IV − III
J I − IV ⎤
J II − IV ⎥⎥
J III − IV ⎥
⎥
J IV − IV ⎦
J I − IV = 0, J II − III = 0
J III − II = 0, J IV − I = 0
Area III
Area IV
z By overlapping decomposition of J with tieline buses overlapped, one can effectively get
a probably promising decentralized indicator
of static voltage severity
1/10/2006
Ilic & Xie, ECE/CMU
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Example: 7 Bus System
All intra-area transmission lines are with impedance of 0+j0.1
Bus 1 is a slack bus, bus 2, 5, 7 are P-V buses, and bus 3,4,6 are
P-Q buses.
Load at bus 3, 4 and 6 have same power factor of 0.995
Keep increasing load at bus 6, while keeping all the other loads
constant
1/10/2006
Ilic & Xie, ECE/CMU
5
Simulation Results
z By disjointly partitioning load flow Jacobian
Load level (load at bus 6)
Minimum
eigenvalue of
system load
flow Jacobian
Minimum eigenvalue of
area-based partitioned
block matrices
250 + j25 MVA
0.4931
2.0686
285 + j28.5 MVA (close to
collapse)
0.2222
1.1355
z By overlapping decomposition of load flow Jacobian
Load level (load at bus 6)
Minimum
eigenvalue of
system load
flow Jacobian
Minimum eigenvalue of
overlapping
decomposed block
matrices
250 + j25 MVA
0.4931
0.4931
285 + j28.5 MVA (close to
collapse)
0.2222
0.2222
1/10/2006
Ilic & Xie, ECE/CMU
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Observation
z Simple block partition of system load flow
Jacobian does not effectively indicate static
voltage severity
z By overlapping decomposition of load flow
Jacobian, the minimum eigenvalue of block
matrices can serve as an indicator of system
static voltage severity
z Only tie line buses’ entries in system load flow
Jacobian matrix are needed to exchange
between neighboring areas
1/10/2006
Ilic & Xie, ECE/CMU
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Conclusion
z Our work proposes a possibly promising
indicator for static voltage stability
z This indicator could be achieved in a
decentralized way. Only a small piece of
information is needed to exchange
between neighboring areas
z More theoretical work is required for
justification of guaranteed performance
of this method
1/10/2006
Ilic & Xie, ECE/CMU
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Key References
z [1] P.W. Sauer, M.A. Pai, “Power system
steady-state stability and the load-flow
Jacobian”, IEEE Transactions on Power
Systems 1990
z [2] M. Ilic, E. Allen et. al, “Preventing
Future Blackouts by Means of Enhanced
Electric Power Systems Control:
From Complexity to Order”, IEEE
Proceedings 2005, pp 1920-1941
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Ilic & Xie, ECE/CMU
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Thank You!
1/10/2006
Ilic & Xie, ECE/CMU
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